In Exercises 51 - 54 , complete parts ( a ) , ( b) , and ( c...

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

$$c = 1 , f ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { x - 1 } , } & { x < 1 } \\ { x ^ { 3 } - 2 x + 5 , } & { x \geq 1 } \end{array} \right.$$

Key Concepts

Evaluating Piecewise Functions

Evaluating a piecewise function requires determining which part of the definition applies based on the input value. Knowing how to correctly choose the appropriate expression for evaluation is a fundamental skill when working with these functions, as it ensures accurate computation for any given input.

Continuity at Boundary Points

Continuity is an important concept in analyzing piecewise functions, especially at the points where the definition changes. To determine if a function is continuous at a boundary, one must check that the limit from the left equals the limit from the right and that they both equal the function value at that point. This concept is essential in many areas of calculus and helps in understanding the smoothness of the function.

Piecewise-Defined Functions

A piecewise-defined function is a function described by different expressions depending on the interval of the independent variable. This type of function is common when different rules apply to different parts of the domain, and understanding how and why the function is defined in separate parts is essential in analyzing its overall behavior.

Domain Restrictions and Conditions

When dealing with piecewise-defined functions, it's important to note the conditions or restrictions on the domain that dictate which expression is used for a given input. These conditions ensure that each part of the function is applied appropriately, and they are critical in understanding the full behavior of the function and avoiding possible undefined expressions.

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